Tangent Line Equation Calculator
Find the equation of the tangent line to f(x) = axⁿ at any point x₀.
Returns the slope, y-intercept, and full equation in slope-intercept and point-slope form.
The tangent line to a curve at a point touches the curve at exactly that point and has the same slope as the curve there. The slope at any point is the derivative.
For f(x) = a * x^n, the derivative is f’(x) = n * a * x^(n-1).
At the point x0, the slope of the tangent is m = f’(x0) = n * a * x0^(n-1). The point of tangency is (x0, f(x0)).
Point-slope form: y - f(x0) = m * (x - x0) Slope-intercept form: y = m * x + (f(x0) - m * x0)
The y-intercept is b = f(x0) - m * x0.
One thing students often confuse: the tangent line is a global linear function, not just a local approximation. It extends to the full plane as a straight line. The fact that it matches the curve at x0 (same position and same slope) is what makes it “tangent.” For higher derivatives, you can build a Taylor polynomial that also matches curvature – but the tangent line only matches to first order.
The tangent line is also the best linear approximation to f near x0. If you need to estimate f(x0 + small delta) without computing f directly, the tangent gives: f(x0 + delta) ≈ f(x0) + f’(x0) * delta. This is linear approximation, widely used in engineering and physics.